Do the critical (2,2) strings know about a supergravity in 2+10 dimensions ?

The effective field equations of motion for a mixed theory of open and closed (2,2) world-sheet supersymmetric critical strings are shown to be integrable in the case of an abelian gauge group. The Born-Infeld-type effective action in 2+2 dimensions is intrinsically non-covariant, and it can be interpreted as (a part of) the F-brane world-volume action. The covariant F-brane action is unambiguously restored by its maximal (N=8) world-volume supersymmetry. The 32 supercharges, the local SO(2,1) x SO(8) and rigid SL(2,R) symmetries of the F-brane action naturally suggest its interpretation as the hypothetical (non-covariant) self-dual `heterotic' (1,0) supergravity in 2+10 dimensions.